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Properties

Label 2-950-95.49-c1-0-3
Degree $2$
Conductor $950$
Sign $0.466 - 0.884i$
Analytic cond. $7.58578$
Root an. cond. $2.75423$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.866 − 0.5i)2-s + (−0.866 − 0.5i)3-s + (0.499 + 0.866i)4-s + (0.499 + 0.866i)6-s + 4i·7-s − 0.999i·8-s + (−1 − 1.73i)9-s + 3·11-s − 0.999i·12-s + (1.73 − i)13-s + (2 − 3.46i)14-s + (−0.5 + 0.866i)16-s + (−5.19 − 3i)17-s + 2i·18-s + (3.5 + 2.59i)19-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (−0.499 − 0.288i)3-s + (0.249 + 0.433i)4-s + (0.204 + 0.353i)6-s + 1.51i·7-s − 0.353i·8-s + (−0.333 − 0.577i)9-s + 0.904·11-s − 0.288i·12-s + (0.480 − 0.277i)13-s + (0.534 − 0.925i)14-s + (−0.125 + 0.216i)16-s + (−1.26 − 0.727i)17-s + 0.471i·18-s + (0.802 + 0.596i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.466 - 0.884i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.466 - 0.884i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $0.466 - 0.884i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ 0.466 - 0.884i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.634675 + 0.382663i\)
\(L(\frac12)\) \(\approx\) \(0.634675 + 0.382663i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 + 0.5i)T \)
5 \( 1 \)
19 \( 1 + (-3.5 - 2.59i)T \)
good3 \( 1 + (0.866 + 0.5i)T + (1.5 + 2.59i)T^{2} \)
7 \( 1 - 4iT - 7T^{2} \)
11 \( 1 - 3T + 11T^{2} \)
13 \( 1 + (-1.73 + i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (5.19 + 3i)T + (8.5 + 14.7i)T^{2} \)
23 \( 1 + (5.19 - 3i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 2T + 31T^{2} \)
37 \( 1 - 10iT - 37T^{2} \)
41 \( 1 + (4.5 - 7.79i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.46 - 2i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-5.19 + 3i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (4.5 - 7.79i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2 - 3.46i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6.06 + 3.5i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-3 + 5.19i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-0.866 - 0.5i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (2 - 3.46i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 3iT - 83T^{2} \)
89 \( 1 + (-3 - 5.19i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-14.7 - 8.5i)T + (48.5 + 84.0i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.02153335175736849314243994800, −9.253883584991822195300021768731, −8.745830232621326750504714274519, −7.86286925961538789347959929882, −6.56491162875177472362685968503, −6.13753774016493014430685776534, −5.10107888811314782736796660770, −3.64398279032656960535210406056, −2.58504170256476707877433006267, −1.29940838718436080897476112136, 0.49204898558125110267532481645, 1.99123612801223462201687630043, 3.85084854726474291964734602701, 4.51059847412809537770235779511, 5.74695227921424646434916344299, 6.62718342087125313502679838885, 7.27331410165023910476046279217, 8.258041408658967622893955828608, 9.036013554772395989147657355060, 9.999938917270649970287659278206

Graph of the $Z$-function along the critical line